text:
If a balloon lies within gun range, why would a 21st century warplane shoot it with a missile, instead of rotary guns?
I am not necessarily referring to current events, but you can if you want. Presuppose a balloon is within altitude, and range, of the guns of a warplane made in the 2000's.
Particularly if a military wants to salvage and deconstruct a balloon, then why not use bullets? Leaving something behind to be analyzed! An air-to-air missile is both more expensive and leaves less to be salvaged!
On August 29, 1998, Canadian Major Roland Lavoie remarked, "Also, it might be overkill spending a couple of hundred thousand dollars on a missile to shoot down a balloon that's drifting away."

different but related: What other times has a Chinese high-altitude balloon entered US airspace? and To what extent is modern "maneuverable spy balloonery" really a thing?

Maybe I'm thinking too simplistic here - but the balloon is pretty much stationary compared to the high speed of the war plane. I'd simply assume that the max firing range of the guns will pose a danger to the plane. You fire about a mile but that mile is flown in mere seconds - and the balloon is still "there", ready to colide with your plain .. Use a missile over like 10 miles and you have ample time to avoid the balloon obstacle

We were speculating that we got the benefit of a live missile test. Drones are expensive. This target didn't cost us anything.

And some fighter pilot gets to paint a little balloon just below his/her canopy! Take that Tom Cruise...

Some factors that do not seem to be discussed in the other answers:
The Balloon was estimated to be around 60 to 65,000 feet, which is approximately the service ceiling of the F-22, the US fighter with the highest capabilities.
This poses several serious problems for the fighter and its gun.

For an F-22 to maintain flight at that extreme altitude, where the air is very thin, it needs to be supersonic to generate enough lift.  When going supersonic, there is a good chance that the plane will actually overtake its own bullets!  The bullets start to slow down as soon as they leave the barrel due to the shockwave in front of them, even as the plane is keeping its speed up.  It would really be bad if our plane ended up shooting itself down.  There is one confirmed case of a plane shooting itself down with its own gun, and I've heard stories of two others (although I cannot find documentation to back it up).

Because the plane would be flying at super-sonic speeds, and the balloon is virtually stationary, it might be extremely difficult to fire accurately at a target as the pilot passes it at extremely high speed.

It is entirely possible that the balloon might've been above, even well above the plane.  The plane might not be able to get a nose-up attitude to shoot at the balloon when at the edge of its service-ceiling.  This is supported by reporting from TheDrive, indicating the balloon may have been 7,000 ft above the plane.

An F-22 Raptor from the 1st Fighter Wing at Langley Air Force Base fired a single AIM-9X Sidewinder missile that downed the balloon from an altitude of 58,000 feet. The balloon was as high as 65,000 feet.

Shooting a missile covers all these problems:

The missile has its own engine and will not be overtaken by a supersonic jet.
The missile has its own guidance, and would not rely on the pilot firing guns at exactly the right moment while at supersonic speeds.
The missile can gain altitude and go even higher than the plane that fired it.

Self-shootdown is unlikely. At high altitude, bullets have a lot less drag as well. The main advantage of the missile is that it packs a lot of explosives, so can do a lot of damage quick.

The narration in this video also supports the idea that the balloon was well above the F-22-- the figures given are 58,000' for the jet and "between 60 and 65 thousand feet" for the balloon.  It looks closer in the video at first glance, but it's hard to judge at that distance.  Factoring in the idea that fighter pilot would not want to be too close to the exploding warhead, a large vertical separation seems plausible -- https://www.youtube.com/watch?v=h7_K26Eg1UI

The bullets may start slowing down as soon as they leave the barrel, but they don't generate lift, so they start falling to earth due to gravity. I find it hard to believe that a supersonic plane is going to run into its own bullets and shoot itself down.

@Mohair If the bullets are fired upwards, they won't start falling immediately; they will in fact pass back through their starting altitude. Plus the jet is near its altitude ceiling, so is probably going to want to descend. This post links to a page about the time an F-11 shot itself down, and it's a very expensive and dangerous mistake.

https://www.facebook.com/reel/565885115598926?s=yWDuG2&fs=e&mibextid=Nif5oz

Gail Helt, a former CIA analyst suggests that due to the huge size of the balloon (and almost zero pressure difference between inside and outside) that shooting the balloon full of bullet holes would not bring it down quickly. It would just continue to drift very slowly to lower and lower altitudes (right into commercial aviation traffic) before landing.
From DW News' February 4, 2023 China's spy balloons: What's the geopolitical fallout?

These balloons are so huge that even that kind of - that number of ammunitions - that amount of ammunition set towards it is only going to cause a very very slow leak, and it's not going to come down immediately.
You can't just pop the balloon and it's going to go away.

Presumably the balloon's payload is both of great interest and deserving of some caution. It could have defenses, a self-destruct mechanism, or other undesirable materials so one might not want to simply destroy the payload with a missile, and yet one might not want it landing on the population either.
Thus once it's over the Atlantic an air recovery operation might occur, or now that the balloon is having its "15 minutes of fame" US authorities might even just let it continue and irritate/traumatize other countries for political benefit.
update 1: Then again, maybe not: Video appears to show suspected Chinese spy balloon being shot down reportedly an F-22 fired a single AIM-9 Sidewinder missile and apparently the blast fragmentation pattern from a proximity detonation was enough to rip the balloon wide open and allow the payload to fall into the ocean, and a recovery mission is on its way.
Hat tip to @tedder42's comment:

The Drive suspects the AIM9 was used because it has a laser fuze, not radar

Nice real-time video at Fox 13 Seattle's The moment the Chinese spy balloon was shot down, ordered by President Biden where the balloon is shredded.

Source: New York Times February 5, 2023 China Condemns U.S. Decision to Shoot Down Spy Balloon footage credited to Associated Press
update 2: From BBC's February 8, 2023 When a Canadian weather balloon went rogue:

"We asked the Canadian military if they would consider shooting it down, sending somebody out for target practice," Mr Sommerfeldt said.
A fighter jet had a ceiling of about 60,000 feet, he recalled and "the balloon, of course, was a lot higher than that".
Hoping to save the research, he asked them to shoot upwards and aim for the suspension point, where the release mechanism was for the package of scientific instruments, which were attached to a parachute.
According to news reports from the time, the Canadian CF-18s fired more than 1,000 rounds of ammunition at the balloon - the pilots refrained from using air-to-air missiles.
The volleys of ammo did not work.
"It wasn't enough to shake loose the release mechanism. They probably hit the balloon too. But those small bullet holes and a balloon that size would have almost no effect," Mr Sommerfeldt said.

and

It then drifted back towards Norway before finally landing on Finland's Mariehamn Island after what the Tribune News Service dubbed a "nine-day odyssey".
The instrumentation was sent back to Canada and reused (though there were some bullet holes on the instrument package and its parachute). Mr Sommerfeldt said.

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Huh, I thought the AIM-9X had a radio fuze.

One reason that likely factors into this seemingly odd decision is that Canada tried and failed to shoot down one of their wayward weather balloons using only guns in 1998

A runaway weather balloon floating toward Britain over the North Atlantic is proving a tough target for some of Canada’s top guns.
Jet fighters trying to bring the balloon down fired more than 1,000 rounds into it Thursday, but it remained aloft. The air force hopes the now-leaking balloon will eventually come down.
The CF-18 fighters that caught up with the balloon over Newfoundland and failed to bring it down were equipped with air-to-air missiles, but Lavoie said the pilots refrained from using heavier firepower.
"Citizens would not have appreciated having a missile blowing over their heads,″ he said. "Also, it might be overkill _ spending a couple of hundred thousand dollars on a missile to shoot down a balloon that’s drifting away.″

There's also the fact that nobody was worried about a Canadian balloon drifting over the UK (they used to be the same country). The US would clearly like to know what the Chinese were getting with this balloon. A guided missile with a touchy detonation trigger likely was a better bet.

Also, it needed to come down in a relatively narrow 12 mile space between the coast and international waters. Just punching holes in the balloon at 65,000 feet may have let it keep drifting, and gone past the 12 mile limit. They needed to destroy the balloon, and let the package come pretty much straight down.

@WPNSGuy, is the 12 mile limit relevant?  I thought it would be finders keepers in international waters.

I don't know for certain, but if it came down outside the 12 mile, China could claim the US shot down a peaceful aircraft in international airspace. Just a possibility, but I'm sure the legalities were discussed in depth over the last few days.

Probably yes, although the ADIZ probably goes much farther than that. But also the retrieval of the debris is more difficult the farther (or even deeper) it is in the ocean.

It's also a good idea to know if a certain munition is capable of locking on to a certain target type and destroying it. This was a valuable experiment in that context, and was funded by the Chinese defense establishment.

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As others have mentioned, there is little appreciable pressure between the outside and inside of the balloon meaning it wouldn't "pop" if you poke it full of holes. To give an idea of how many holes we're talking about, time to do a little math.
The chinese balloon is reported to be about 200 feet tall which is gives it a radius of roughly 30 meters assuming its a sphere. Giving a surface area of approximately 11,309 square meters.

The F-22 has 480 rounds of 20 mm ammo, and since its unlikely the bullet will detonate hitting a thing skin fabric of the balloon, we're at best getting 2 holes (entry and exit). The area of each hole will be a circle 314 mm^2 or 0.000314 m^2. 480 * 2 * 0.000314 would mean you'd poke about 0.301 square meters if you get 100% hit rate.

A real world equivalent would be roughly a hole the size of a basketball (24 in wide) in object the size of a football field (200 feet wide). It will leak very very very slowly.

+1 btw there's some goodies for leak rate estimation in this answer to How could the 2018-08-30 Soyuz MS-09 / ISS leak be so slow?

Here's a nice calculation:
Its been reported that the pressure differential of the balloon is maybe only 10% higher than the surrounding environment. The pressure at 65,000 feet is 0.82 PSI, so we can assume the balloon pressure is at 0.902 PSI. For a diameter of 24.37 in
https://www.tlv.com/global/TI/calculator/water-flow-rate-through-orifice.html?advanced=off
This gives us a leak rate of 805.4 m^3 / hour. The ballon has a volume of 113,097 m^3 so it would take 140 hours or about 5.8 days to deflate if you fired a full volley of 20mm rounds.

Actually I might have made a mistake with the wrong calculation and forgot to take into account temperature, and use a water vs. air flow rate calculator.
https://www.tlv.com/global/TI/calculator/air-flow-rate-through-orifice.html
Using that calculator and -56.5 C (216.5 K) which is the temperature at 65,000 feet, we get a flow rate of 6303 Nm^3/hour which is still about ~18 hours.
I might be using the equation wrong since the calculate returns "normal M^3 / hour" which normalized the flow rate at 0 degree C and 1 atm of pressure.

Good answer. Just a reminder: your calculation for the area of bullet hole is for when it hits exactly in the centre. If the bullet hits from an angle, the hole it creates will have a greater area than the bullet cross-section.

Aside from the (good) answer that, because of lifting gas at the same pressure might only slowly vent through even numerous bullet holes, leading to a potentially slow uncontrolled descent through air traffic lanes, one might be a bit disappointed by a lack of creativity in the response to a drifting balloon.

why not capture it with another balloon?

A rigid airship, like the Zeppelin/Goodyear blimps could be modified for stratospheric flight with external expanding balloons.  Pilots would wear SR-71/astronaut type suits.  Balloons have flown to over 120,000 feet.  This one was sailing along at half that height.
The dirigible could even be remotely controlled.  It would only have to latch on to the wayward balloon, release its own lifting gas, and drag it down.
Inspecting and returning it relatively intact to its owners with a mild admonishment of "please be more careful next tine" would seem far more appropriate and diplomatic instead of blasting it out of the sky with a missle.

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authors: Vladimir F Героям слава


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In The Final Countdown, why did F-14 gun down the first A6M2 Zero, but the other F-14 fired an air-to-air missile at the second A6M2?
It feels contradictory for one of the F-14's to gun down the first A6M2 Zero, but the other F-14  to fire  an air-to-air missile ("AAM") at the second A6M2. Is this distinction a goof? If not, what distinguishes the choice of different weapons?
Did the first pilot use rounds from the M61A1 Vulcan rotary cannon, because he needed merely a handful and they are cheaper?  I know from military service that militaries don't want you to shoot rabbits with elephant guns, or kill flies with cruise missiles. On August 29, 1998, Canadian Major Roland Lavoie remarked, "Also, it might be overkill spending a couple of hundred thousand dollars on a missile to shoot down a balloon that's drifting away." I can't identify the air-to-air missile ("AAM") from this Youtube video, but it costs more than the handful of rounds from the F-14's rotary cannon.

You might want to edit your question from, "why did the first pilot..." to something more like "why did the scriptwriter..." chose this or that.  I would seriously doubt if the pilots flying those F-14's for the cameras had any input at all into the choice of weapons that made the movie screen.

YouTuber and former F-14 RIO Ward Carroll published a very thorough video commentary on this film. It's well worth watching. There's tons of interesting information in it (for example, and since you wondered, the missile was an AIM-9 Sidewinder) but nothing is said about the guns vs. missiles.
In addition to the obvious answer "because that's how they wrote the script," I suspect one important factor was that they wanted to showcase all the capabilities of the F-14. Much of the film is in fact a "personal ad" for U.S. naval aviation.

In real life the weapon of choice is dictated, in simple terms, by distance to target and type of target (air to air, air to ground, air to radar etc.).
While the effective range of aircraft gun is very long (see "The fighter plane that shot itself down"), the accuracy falls off very rapidly. Using a gun at targets several hundreds of yards away becomes essentially a game of luck if the target is maneuvering.
Missiles, on the other hand, need to acquire a target lock, which may be somewhere between difficult to impossible if a maneuvering target is very close. A missile may also have difficulties maintaining target lock after launch towards a close target.

Can't wach that video, as I am not a member of PopMech... there are other youtube videos about it, could you consider a replacement link?

source: Stack Exchange

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Can I takeoff as VFR from class G with 2sm vis. even if that's IFR in the categorical outlooks?
Weather minimums for class G during the day (below 1200'AGL) are 1SM vis. and clear of clouds.
The categorical outlook (AIM 7-1-7) for IFR is 1sm to 3sm visibility and a ceiling of 500'AGL to 1000'AGL.
Now if I want to takeoff from class G airspace during the day as VFR with 2SM of visibility and a ceiling of 700'AGL, am I allowed to do it?

I think you are mixing up ceilings and cloud coverage.
14 CFR 91.155(a) says how much visibility you must have, and how far away from clouds you must be in VFR. In class G at day under 1200 feet AGL, this is 1 mile visibility and "clear of clouds."
14 CFR 91.155(c) is where "ceilings" come in:

Except as provided in § 91.157, no person may operate an aircraft beneath the ceiling under VFR within the lateral boundaries of controlled airspace designated to the surface for an airport when the ceiling is less than 1,000 feet.

A class G airport does not have controlled airspace designated to its surface, so this does not apply, and you can operate under lower ceilings as long as you can maintain the appropriate visibility cloud clearance requirements.

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Lifting Line Theory - The general lift distribution
While studying the derivation of the lifting line theory using different text books like Fundamentals of Aerodynamics by John D. Anderson, Precisely when deriving the general lift distribution over the wing, it assumed the following equation
\begin{equation}
\Gamma(\theta) = 2bV_{\infty} \sum_{n=1}^{N} A_{n} \sin(n\theta)
\end{equation}
How did he just assumed that multiplying the Fourier series expansion by $2bV$ is a good assumption, I mean there must be a mathematical motive to do so, and I could not find anyone explaining the equation, so I hope someone can help me with that.
Thanks in advance
P.S
I suspected that the value  $2bV$ represents the the lift distribution at the origin as in the equation
\begin{equation}
\Gamma(\theta) = \Gamma_{0} \sin \theta
\end{equation}
But still I can't relate both quantities. I thought that I can derive it from the equation
\begin{equation}
\alpha_{i} = - \frac{w}{V_{\infty}} = \frac{\Gamma_{0}}{2bV_{\infty}}
\end{equation}
but to do so I think we should assume that the induced angle of attack is equal to one and I don't understand that physically.
Thank you so much for your help!

2bV is Gamma zero (eq 5.47 in my version)

When I compared both equations I deduced that the quantity is actually gamma zero but I still don't how he got from equation 5.47 to the Fourier expansion series equation.

Your question concerns the lift over span (3D) while my answer was for the 2D case.


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How to calculate sfc per throttle setting of an turbofan engine?
I was looking for sfc per throttle setting of an engine. I was not getting the performance data of that engine. So is there any way to calculate it by formula?

It's going to depend on the load put on the engine.  RPMs alone don't tell anything about it, other than the average torque generated by the engine equals the load resistance at this RPM (or else it speeds up or slows down).

The variation of TSFC vs. Thrust (as you vary throttle) for a jet engine (at a given altitude and airspeed) is usually called a 'thrust hook'.  So you will similarly hear a plot of BSFC vs. Power for a piston engine called a 'power hook'.
Piston engines can be a bit more complex because what happens with the variation of throttle depends on what kind of propeller you have (fixed pitch vs. constant speed).
The (mechanically more complex) constant speed prop is a lot easier to analyze -- you already know the RPM.  Consequently, a curve depicting throttle variation (at a fixed flight condition) can be readily constructed.
For a fixed-pitch prop, the propeller (load) determines the relationship between power and RPM (at a given flight condition) as throttle is varied.  Different props will have different load curves.
Consequently, you need a more complex model for the engine to be able to analyze it connected to an arbitrary fixed-pitch prop.  This model is not just a curve, it is a map.  Google 'bsfc map' and you will find plenty of examples.
Although some engine parameters are easy enough to come by (RPM, bore, stroke, number of cylinders, compression ratio, etc) -- those are not enough to construct an accurate model of engine performance.  To do that, you need to know detailed information about the friction between the rings and cylinders, in the bearings, how the valves are operated, etc.  All of these things contribute to losses that are important for determining the engine's fuel efficiency (and how it varies with speed and power).
All those factors are typically determined empirically (through test) -- not just known from analysis.
So, your best bet is to find some engine performance charts for your engine -- or a similar engine.  What engine(s) are you considering?  Will you be using a fixed pitch or constant speed prop?

You have to know the horsepower being generated at a given throttle setting, which means you need the manifold pressure and RPM, and using those values, derive the horsepower that results from the engine's power chart. So without a power chart, or some other data that tells you what HP is achieved at some throttle opening, you are stuck.
If you know the HP being created, and it's a carbureted air cooled piston engine, you can use an SFC value of 0.45 lbs/hp/hr as a ballpark number. So if you know that the throttle setting is giving you 100 HP, at some manifold pressure and RPM, you can assume the fuel burn is 45 lbs/hr, or 7.5 US Gal/hr. This is about what the cruise fuel consumption is for a 140 hp Lycoming O-290 that makes about 105 hp at 75% cruise.
For a fuel injected engine you can reduce the value to a bit lower, say .42/lb/hp/hr.
Car engines are in the high threes, and diesels are in the low threes. Two strokes and turboprops are thirstier, in the 5-7 lb/hp/hr range.

Yes finger trouble. Thanks for the catch.

My prior answer assumed you were looking at a piston engine.  This answer will reflect your update to interest in a turbofan engine.
You are interested in calculating the thrust hook (TSFC vs. Thrust as a function of throttle) at a given flight condition (Mach and altitude).  Here is an example for a turbofan engine:

As you can see, at each flight condition (numbered line), SFC and Thrust vary along a curve (as you vary throttle).  Note that these hooks do not include the flight idle condition (very low throttle).  Flight idle would show SFC getting substantially worse very quickly.
There is no simple equation to calculate a thrust hook.
Instead, you typically use a computer program capable of performing off-design cycle analysis.  Usually a first undergraduate course in propulsion will discuss on-design cycle analysis (first ideal, then non-ideal).  Off-design cycle analysis is usually not discussed until a second undergraduate course or a graduate course in propulsion.
In on-design cycle analysis, we analyze the engine at a single design point (flight condition and throttle setting).  This analysis allows us to size the engine to meet a design thrust.  Sizing the engine in cycle analysis means calculating the flowpath areas at each station (inlet, compressor inlet, compressor exit, combustor, turbine inlet, turbine exit, nozzle, etc).
In off-design cycle analysis, we take the fixed engine from the on-design analysis (leave the areas and other properties fixed) and then model its operation at different flight conditions and throttle settings.
An off-design cycle analysis requires compressor, fan, and turbine maps.  These are performance models of the turbomachinery components.  I.e. not just an on-design compressor pressure ratio and efficiency, but a map of how compressor pressure ratio and efficiency vary as functions of corrected speed (RPM) and corrected mass flow (compressor face Mach number).
At every operating condition, we solve to find the equilibrium point where the compressor and turbine are balanced (power in equals power out) at the same speed (RPM) and mass flow.  This is an operating point.
At each flight condition, we calculate an 'operating line'.  This is the locus of all the operating points at that flight condition (as we vary throttle).  This operating line is usually plotted on the compressor or fan map, but once you can calculate an operating line, you can also calculate a thrust hook.
If you have thrust hooks at a wide variety of flight conditions, (i.e. a complete performance model of the engine), you have an 'engine deck'.  A deck is the industry term for the data package an engine company would provide to an airframe company to use when designing and analyzing their aircraft.  Back in the day, it was a deck of punch cards.  Today, the 'deck' name persists, but an engine deck is usually supplied as a computer program provided from the engine company to the airplane company.
A bunch of scaling is often performed for engine data.  This isn't exactly non-dimensionalization (as the results are still dimensional quantities), but it works similarly.  Each parameter is 'corrected' in this process.  Mass flow becomes corrected flow.  Thrust becomes corrected thrust.  Compressor and turbine maps are usually provided in corrected form.
With some limitations, an engine's operating line at different flight conditions can fall onto a single operating line when written in corrected properties.  Consequently, if you have a corrected operating line, you may be able to get to a hook at many flight conditions (i.e. a deck).
If you have a corrected operating line, 'dimensionalizing' that data for an engine will be the closest you can come to a simple equation for calculating a thrust hook.  It still isn't easy -- and that line was generated with sophisticated analysis (off-design analysis) that was in turn built on sophisticated models of the turbomachinery (maps).  Neither of those steps submit to simple analysis.


text:
How to plot figure of merit vs blade loading coefficient for helicopter rotor?
I know the formula of figure of merit and blade loading coefficient individually but how the plot can be obtained I am not getting that. What would be the formula for figure of merit in terms of blade loading coefficient. Formulas I know are:

Please edit your question to include which equations you have to start with.  There is nothing we can do to help this question without simply doing your homework for you -- and that is not the purpose of this forum.

@RobMcDonald when the FOM, coefficient of thrust, solidity of the rotor all are fixed which will be the variable in the plot?

My definition of blade loading coefficient is ($C_T/\sigma$).  Plotting FOM vs. $C_T/\sigma$ is a common thing to do.  Typically, the $C_T/\sigma$ variation represents a change in collective for a fixed rotor -- which means that $C_T$ is varying.  Again -- please share the formulas you have and we can help show how to use them - or suggest the missing formula.

What you are missing is the rotor 'polar' equation.  While not exactly the same as an aircraft's drag polar, it plays that role in performance equations.
$C_P=\frac{\kappa {C_T}^{3/2}}{\sqrt{2}}+\frac{\sigma C_{d,0}}{8}$
An ideal rotor has $\kappa$=1 and $C_{d,0}$=0.  Such a rotor has $FM$=1.0.
The $\kappa$ is a non-ideal induced flow factor.  It plays the same role as an Oswald efficiency factor, but is inverted in definition.  A typical value might be $\kappa$=1.15.
The $C_{d,0}$ is the airfoil profile drag coefficient.  It contributes to form the profile drag power.  A typical value is $C_{d,0}$=0.01.
The figure of merit $FM$ is the ratio of the ideal power to actual power.
$FM=\frac{\mathrm{ideal}}{\mathrm{actual}}$
I.e. applying it to our definition above, we get...
$FM=\frac{ \frac{{C_T}^{3/2}}{\sqrt{2}}}{\frac{\kappa {C_T}^{3/2}}{\sqrt{2}}+\frac{\sigma C_{d,0}}{8} }$
Which you should be able to plot.

How to get the value of k?

You perform complex analysis on your rotor design -- including planform, airfoils, twist, etc.  Ideally under loaded conditions as rotors deflect substantially when operating.  Or you measure performance of your rotor and choose k to match the data.  Or you use historical data for similar rotors.  Typical is 1.15 -- 1.0 is ideal, you can't go lower than that.


text:
Can someone recommend a good book/paper on understanding shockwaves over aerofoil?
I’ve been trying to understand shockwaves around an aerofoil from some contour plots I saw here and on the internet, but I’m struggling to understand them
Can someone please recommend a good book on understanding shockwaves (how they form, what they mean, what affects them,…) please?

Please note that resource location is specifically off-topic. If you can reword this to ask about the subject rather than resource location for the subject I don't see why it would be off topic.

Hi, sorry forgot to write shockwaves around aerofoil, then it’s aviation related

It's not that part that makes it off-topic. As I said in my first comment, it's the "resource location" which is off topic. Anyway, looks like you got a good answer, so just for future reference.

Simple shockwaves are best understood starting with (quasi) one-dimensional flow -- flow in a tube.  The quasi allows the area of the tube to vary so long as the variation isn't so rapid that the non-axial components of velocity become very important or the variation of properties across the tube become important.
Rocket nozzles (and jet engines) are a great example where quasi 1D flow assumptions are extremely useful.
Any book, course, or set of notes entitled 'Gas Dynamics' should be a good introduction to one-dimensional compressible flow including shocks.  One popular text is by James John.
Next, you'll be interested in 2D flow -- there the oblique shock (and conical shocks in 2D axisymmetric flow) come into play.  These are also very simple -- you will also learn about expansion fans at this point.  This kind of flow is ideal for understanding diamond-shaped supersonic airfoils.
Most aerodynamics books will cover this kind of flow.  It would typically be covered in a second course in aerodynamics in a university (so you're looking at the second half of the book).
A great resource that can't be skipped is NACA TR-1135 contains all the formulas and charts you could want for elementary compressible flows.  A lot of the textbooks will be dedicated to deriving, explaining, and duplicating what you'll find in 1135.
Once the surfaces are curved -- and/or you are dealing with flow that is not solidly supersonic (i.e. you're in the transonic region).  Then the flow becomes much more complex and there aren't any simple answers anymore.  I would encourage you to check out Bill Mason's (RIP) notes as a nice overview on the application of transonic flow to airplanes.
Unfortunately, I have a feeling that these kinds of flows (with shocks balanced on the top of a transonic airfoil) are what you are interested in.  These are extremely challenging to analyze and design.  Fortunately, having a general idea what is going on is much more achievable.


text:
Disadvantages of the Fowler trailing edge flap?
I get that the trailing edge plain flap has disadvantages (like tendency to separate flow as I read on a question on here)
But what’s so bad with the fowler flap? Aren’t they like really efficient and used on jet aeroplanes?

Yes, flowler flaps (specifically slotted fowler flaps) are used on virtually all airliners because they provide the best efficiency of all the various flap designs.
The major downside is that they are more complex, and therefore expensive to both install and maintain. But in most large aircraft it's a worthwhile investment.
In small aircraft you may run into the problem of fowler flaps being too efficient. They will have less form drag than a plain flap, resulting in a shallower approach, and also after touchdown you may not have enough weight on the wheels for effective braking, without the help of spoilers and reverse thrust.

Their complexity (in addition to making them expensive) also makes them heavy.

Do they still have gaps in between the bits please? I mean like a normal flap does please?

@FlightWatcher -- those "gaps between the bits" are really the basic characteristic of a Fowler flap.  Not sure which gaps you would be referring to in the case of a "normal flap".


text:
Is there any evidence that birds use thermal currents generated from highways to glide?
I am looking for an article on the subject matter of birds using thermal currents generated from highways to glide. I know I have seen and read about this before but cannot find any research on the subject matter.

Just  a random thought.  Birds do use thermal currents to glide.  And highways--especially some highways, namely blacktop highways-- undoubtedly do generate significant thermal currents.  So, it seems the most logical hypothesis would be "yes", and that a "no" hypothesis should not be entertained without some rather compelling proof.  So, with that in mind-- could the question possibly be refined?  For example, do you possibly want to ask whether there is evidence of birds using thermals generated from highways to glide?

I don't see how this is related to Aviation. I think it would be better suited for Biology.SE.

Thanks you @quietflyer. the question should be, 'is there evidence...'

You generally don't get "walls" of rising air from a narrow strip heat source like a highway. Thermals tend to come from single sources, parking lots, dry fields, etc.
You do get a kind of "highway in the sky" on certain days with the right convective conditions, where consistent thermal sources, and a smooth flow of the moving airmass, cause thermals that tend to line up in rows downwind of their sources.
Where the thermals are topping out in cumulous clouds, soaring pilots call them "cloud streets". In a glider you can fly along a street, climbing in thermals and descending between them, without really having to turn in the thermals if the conditions are strong enough.
A bird travelling in a particular direction could also exploit a cloud street if the direction was where they wanted to go. But generally birds that are out thermalling are either hunting or just playing around. Glider pilots fly with them quite a lot.

Re " But generally birds that are out thermalling are either hunting or just playing around." -- except during migration season!  Over one hundred thousand hawks typically pass by in one day during the peak of the fall migration season at one flat-land count station in south Texas... sometimes flying near cloud base and disappearing at times into the bottoms of the clouds...   Plus, consider also that on a windy day, a thermalling hawk has to spend a fair amount of time gliding back into the wind just to "stay local", and cloud streets would seem ideal for that too--

Yeah good point.

You see gulls ridge soaring all day in front of any office building when the wind is going. I'll bet geese take advantage of ridge lift in the Appalachians when migrating south.


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Why would attempting to land using only instruments 'almost certainly' lead to a crash, and why can computers do it safely?
I have no real experience in aviation but I was on the site just out of curiosity. On This question, the top answer states that attempting to land with only the instruments would almost certainly lead to a crash, which surprised me. I understand of course that it would be much more difficult to land without being able to see the airfield, but what factors make such an extreme difference? Also, some computer systems can land a plane, and obviously such computers are using only instruments - what capabilities or information do they have that a pilot doesn't?
As a secondary question, could there be circumstances where it is relatively safe (or at least not catastrophically dangerous) for a human pilot to land using only instruments? For example, would it make a difference if you were landing on a salt flat, completely level and with arbitrary clear space in all directions? Are there certain planes where attempting a manual landing with only instruments might be practical?
Edit: To synthesize/summarize what I understand from the answers so far:

Planes capable of automated landing have additional, high-precision instruments that most planes don't have
Computers are able to continuously monitor many different instrument inputs and respond appropriately to them all, while a human pilot would struggle to manage so many different streams of information simultaneously

The Soviet Buran space shuttle flew in space exactly once, with no one on board, and fully computer controlled flight.  It landed perfectly, even though there was a crosswind at crosswind limits.

The instruments used typically by the pilot of a light single engine aircraft (such as depicted in the cartoon in that question) are either a traditional sixpack, or at best a glass cockpit approaching the complexity of those seen in airliners.
Traditional

Glass

In both these setups what you have is speed, direction and of course altitude, but that altitude relies on the pilot having correctly set the barometric pressure. While accurate (assuming set exactly correctly) I'm not sure you would trust it to the foot/meter and therefore using it to tell you when to flare for landing would be.... error-prone to say the least.
Airliners, some of which do support full autoland, rely on much more sensitive sensors, with multiple redundancy. More exactly, they will use a radar altimeter to tell the computer exactly how high above the ground it is. This, coupled with accurate glide-slope information provided by Instrument Landing Systems (ILS) or GPS, will allow airliners to much more accurately use their multi-redundant computer systems to effect a landing without pilot input.
It should lastly be noted that the sort of complex equipment that can be added to light, single engine, aircraft means that it is possible to have autoland - at least for emergency situations. Garmin have brought such a system to market in the last few years which uses a traditional form of  Intrument Landing System (ILS) not GPS.

@757toga Thanks, answer updated

@757toga Im sure.... do any still use ILS solely today?

@757toga True, and ILS is still the norm today, but at least since 2008 you can also perform autolands with GLS. See e.g. Airbus Fleet Readiness for GBAS/SBAS

Cool, thanks for the knowledge. Have updated again

@757toga For GBAS: very few, that's why ILS is still by far the most common today. In Europe, only Bremen, Frankfurt, Zurich and Malaga have working GBAS (only Frankfurt with Cat II). SBAS autoland is still being certified AFAIK.

If one had a long enough runway, and the air was still and uniform, could one adjust attitude and throttle such that one would descend until one entered ground effect, stabilize in ground effect, and then reduce speed until touchdown?  Such an approach might require an order of magnitude more runway than a conventional approach, but some planes can land easily on a runway less than 1/10 as long as some commercial runways.

The radar altimeter is the key. A modern GPS can bring the plane down on the runway, but it can't flare at the end. The datum error (correcting for the earth's actual shape rather than ideal shape) is about 10 feet, which means you will hit the runway. I can't look up the vertical error though.

There are two key factors to consider: precision, and data processing.
The instruments in the cockpit are only designed to get the pilot "close enough" so that when the runway becomes visible they can land visually. One big challenge is judging the flare so the rate of descent is reduced and the airplane touches down at a reasonable rate. Too early and it could stall and hit the ground too hard, too late and it doesn't slow down enough before hitting the ground. This is harder to do only on instruments, especially in most small planes that don't have a radar altimeter to more precisely measure the height above the ground.
Humans have a hard time focusing on more than one thing at a time and it takes some time to switch their focus. Flying on instruments requires a pilot to be monitoring and controlling attitude, airspeed, altitude, and heading all at the same time. In the air they learn to scan the instruments one at a time and this is good enough for the speed at which things happen there, but at landing things must happen much faster. Even a slow computer by modern standards can switch focus millions of times every second. This lets them constantly monitor all of the sensors and process that information to guide the airplane onto the runway.
To put it another way, humans are most efficient processing visual information and that's why they land looking at the runway out the window. Computers are very efficient at processing numbers like acceleration measurements and radio signals, so they can use that information effectively to control a plane even at landing. Computers are also getting better at processing visual information.

+1. If the flight computers were designed to use their synthesized data to draw a 3d view of the world on a screen or VR headset, the pilot could definitely land "on instruments." There are systems that do exactly this, but to my knowledge they haven't yet been used for zero-visibility manual landings. Honestly in this regime, might as well let the computer land. It has fewer points of failure and the technology has been deployed since the 1970s.

Exampled of a 3D HUD on the Space Shuttle: https://youtu.be/xk2rEnxD59s?t=236

Some planes do have autopilots systems that can land the plane. They know as category 3 auto landing autopilots. to use them the plane has to be certified, testing the autopilot system. the crew has to be certified, training them to use it and know its limitations. Finally, the airport has to be certified. The runway requires precision equipment to send the signals the planes uses to know exactly where the runway is.
Some airports like San Francisco have many times each year where the visibility is 0. Only category 3 auto landing systems can be used. Other airports like Phoenix seldom had low visibility. They would not bother to install the equipment needed.
These systems are expensive and few if any private aircraft will be equipped with them. they are normally only found on large commercial aircraft.

In 1929, Jimmy Doolittle became the first pilot to take off, fly and land an airplane using instruments alone, without a view outside the cockpit. Hence a human can do. There there no computers as such in 1929. About ten years will pass till them appearing even on submarines.
However he used "the most heavily instrumented airplane in the world at the time", including a specialized, very precise altimeter and radio marker beacons.
Hence I think typical instruments as installed in that small aircraft may not be good enough or not optimized for the task, even today.


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Can I fly NYC Hudson River SFRA without ADS-B?
My Canadian PA28 doesn't have ADS-B. If I fly around and below A/B/C airspace, can I fly the NYC Hudson River SFRA? Or must I fill the ADAPT authorization form?

Good news, no A airspace near NYC that you'll have to worry about.

Assuming that your aircraft is certificated with an electrical system, no, you cannot fly the Hudson River SFRA unless you have either an approved ADS-B installation or a waiver/authorization from the ADAPT site.
14 CFR 91.225(d):

After January 1, 2020, [...] unless otherwise authorized by ATC, no person may operate an aircraft in the following airspace unless the aircraft has equipment installed that meets the requirements in paragraph (b) of this section:
(d) Except as provided for in paragraph (e) of this section, within 30 nautical miles of an airport listed in appendix D, section 1 to this part from the surface upward to 10,000 feet MSL;

Appendix D, section 1 is the list of "Mode C veil" airports, and includes the three NYC-area Class B airports. The Hudson River SFRA is obviously well within that 30-mile area.
The exception from above is 14 CFR 91.225(e):

The requirements of paragraph (b) of this section do not apply to any aircraft that was not originally certificated with an electrical system, or that has not subsequently been certified with such a system installed, including balloons and gliders. These aircraft may conduct operations without ADS–B Out in the airspace specified in paragraphs (d)(2) and (d)(4) of this section.

(If you are exercising the privilege allowed by paragraph (e) you must remain outside of Class B/C airspace, and below the ceiling altitude of the B/C airspace, or below 10,000', whichever is lower.)

Thanks for you answer, that's what I feared, even if 14 CFR 93.350 calls it a Class B airspace EXCLUSION. Also, the FAA kneeboard" publication specifies that the skyline route (1300'+) is Class B, but doesn't say for the Hudson Exclusion (-1300').

The Class B exclusion is truly a Class B exclusion. That airspace is Class G from the surface up to but not including 700' AGL, and then Class E up to but not including 1300' MSL. But that doesn't matter for your question—the airspace is still within the 30-NM Mode C veil and therefore ADS-B Out is required from the surface up to 10k MSL. (And also required in the Class E airspace at-and-above 10k MSL, regardless of the Mode C veil.)


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Understanding how Attitude Indicator works?

Hi guys. The only doubt I have is if the roll and pitch gimbals shown on the top rotate or remain rigid in space holding the gyro when the airplane is pitched and/or banked. I think that they remain in the same position and the only thing that moves is the instrument case as shown on the bottom picture, what do you guys think?
Thank you

The extremely fast spinning gyro remains in the same orientation in space as the instrument case moves around along with the instrument panel and the whole aircraft. The difference between the orientation of the gyroscope and the casing is mechanically or electronically transformed to display the orientation of the plane
The stabile orientation of the gyro is achieved by the gimbals: each axis has its own gimbal that will allow (almost) frictionless movement with respect to that axis. However the gimbals are actually not free to do what they want so to speak. They are dependent on each other by stacking: each axis/gimbal is dependent on a higher level one that it attaches to. This will in some instances lead into an interesting situation called gimball lock. This video describes the phenomenon:

I think that they remain in the same position and the only thing that moves is the instrument case as shown on the bottom picture...

The instrument reading is directly derived from a gyroscopically-leveled, essentially precession-free, gimbal-mounted sensor.  There is no binding of the freely moving instrument sensor within the instrument case.  This excellent video by Ryan Anderson explains clearly the operation of the freely-moving instrument sensor.  Take a look, one should have no doubts...


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Human error analysis in general aviation
can anybody help me with understanding more about the human factors analysis in general aviation Controlled flight into terrain. I am researching on the topic and I would really appreciate any help in getting further insights.
Edit 1:
As a part of my research on "Designing a sensory warning system to reduce General Aviation Controlled Flight into Terrain". I need to understand the following factors for a pilot during the process of flying:

Mental workload
Decision making
Skilled performance
Human-computer interaction
Human reliability
Work stress and training

I need to understand the above points with a behavioral and cognitive ergonomics point of view.

Thanks a lot for sharing this.  I am looking forward to understanding the pilot's first hand perspective in such situations. I am designing a sensory warning system that would act as a fail-safe measure in case of interpretation errors of warning signs or wrong actions. Would really appreciate for any leads. Thanks again.

@SoumyajitMaiti Welcome to Aviation.SE! Your question is a bit vague at the moment. Can you please edit it to include all details from your comment and anything else that may help to provide a more specific answer?

@Bianfable I can expalin in detail in case you would like to get in touch. I am looking forward to get some qualitative insights rather than quantitative insights, that would help me in better understanding.


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What are the modulation parameters of a DME?
I am having difficulties finding specific information about DME pulse pair modulation. As far as I understand, the gaussian shaped pulse pairs are modulated using amplitude modulation. The carrier wave frequency is equal to the frequency of the interrogation or reply frequency of the channel. It seems the modulation index is 100%.
Example: NAV 115.4 MHz, DME channel 101X, interrogation frequency 1125 MHz and reply frequency 1188 MHz. Aircraft sends a pulse pair to ground station, using carrier frequency of 1125 MHz. Reply from ground station after induced delay with carrier frequency 1188 MHz.
At the receiver they seem to use a envelope detector to retreive the original pulse shape for validation. Am I missing anything? There is barely any info to be found about the modulation.
I have found the following picture:
https://www.researchgate.net/publication/307844436/figure/fig1/AS:403998101262340@1473332061273/DME-pulse-pair-Fisher-2004.png
Thanks!

DME principle

Source.
The interrogator sends a pair of pulses, the DME transponder sends back identical pulses after a fixed delay of 50 μs for X channels, 56 μs for Y channels. The interrogator is able to determine the distance from the round-trip time.
Pulses
The pulses go by pairs spaced by 12 μs for X channels, 36 μs for Y channels.
Rectangular signals generate an infinity of harmonics in their spectrum. To limit the bandwidth to the channel width (1 MHz) a sinc-like shaper is used.
Pulses are not modulated if we except the shaper, it's only CW.
Pulse train with jitter for correlation
The principle is the interrogator repeats pulse pairs at some rate and the transponder replies with the same spacing. The interrogator slightly varies its rate (random jitter) in order to be able to correlate the reply train with its own train and determine the round-trip time, hence the range.
Interrogation rates are:

Faster when trying to correlate the transponder train (about 130 pp/s). The window for correlation is defined by the maximum range of the DME.

Slower when correlated (about 25 pp/s). The window for correlation is defined by the current range of the interrogator and the velocity vector.

A 6-second memory is used if the correlation is lost (extrapolating or not the last known value with the aircraft velocity vector)
Squitter pulses
When there is only a few interrogations, the transponder still maintains a rate of reply pairs of about 2700 pp/s by inserting dummy (squitter) pairs, in order to allow interrogators to identify it and adjust the gain of their receiver.
ID
ID pulses are added every 30/40 s, this is done by using a pulse pairs train at 100 μs spacing to key the Morse signs at 1350 Hz (740 μs spacing).
TACAN / VORTAC DME
In a TACAN the DME reply pairs from the transponder are also used for bearing determination. A rotating directional radiation pattern is used, either using a rotating antenna or a circular phased array.
The pattern rotation rate is 15 Hz, and a ripple is added at 135 Hz for a more accurate bearing determination. The signal strength variation creates at the receiver, like in the conventional VOR, an AM envelope. The apparent modulation index is about 0.2, but is actually dependent on antenna used, it's the result of the minimum and maximum gain in the radiation pattern.
North reference bursts (NRB, 15 Hz) and Auxiliary reference bursts (ARB, 135 Hz) are added to indicate when the rotating pattern has the 15 Hz and 135 Hz maximums facing North.
The composite signal sent, for an X channel is:

Source.
The envelope created by the rotating field can be detected by the receiver to extract the 15 Hz and 135 Hz components and determine their phases using NRB and ARB (135 Hz ripple removed for clarity):

Source.
For a separate DME (not part of a TACAN, e.g. VOR-DME) there is no rotating antenna, no NRB/ARB, the signal amplitude is constant.
More
DME specifications can be found in ICAO Annex 10, volume 1, at §3.5, in particular the shape of the pulses (e.g. rise/fall times).
You can get more technical elements in Test of DME/TACAN Transponders, an application note from R&S.
See also: How does TACAN work? (last section).


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Is the "deaths per passenger mile" metric more useful for evaluating air travel safety than "deaths per flight"?
Commercial airtravel is often called the safest mode of transportation, based on comparing deaths per passenger mile. However, I'm wondering if, among all possible metrics, passenger miles are the best possible metric for airflight safety, to even allow meaningful comparison to the safety of, for example, car travel. By "best possible metric" I mean the metric that is most predictive of airtravel accidents/fatalities of the industry as a whole, not any individual flight.
According to this statistic, the overwhelming majority (79%) of all accidents happen while landing, takeoff, approach, initial climb, and taxi. These are all phases that every flight, whether long or short, have in common. Less than 5% of accidents happen during cruise, the phase in which most of the distance of a flight is covered.
This statistic says that, by my calculation, only about 30% of fatalities between 2011 and 2020 happened during cruise, so 70% are practically independent of passenger miles. These Airbus accident statistics for 2003-2023 broadly agree in terms of the ratio of fatal flight accidents by flight phase.
Based on that, would "deaths per flight" not be a much better metric to evaluate risk, because it is much more predictive of accident/fatality than passenger miles? And yes, I'm aware that you need a common metric to compare airplanes to other transportation, but what's the use of that comparison if that metric barely applies to airplanes to begin with?

How do you define “correct”? Seems this is entirely subjective.

Correct means: which metric gives me the most useful information about the risk of airflight, so I can compare it to other modes of transportation, if possible.  While metrics are indeed subjective and arbitrary, they are not equally useful. "Fatalities by airplane color" does not seem useful, "fatalities per passenger mile" is much more useful, and, which my question is about, "fatalities per flight" might be even more useful because it more accurately reflects the distribution of risks of airflight.

I think the best metric is to ask life insurance (data): probability of a person to die in a car accident (driver, passengers, people outside), flight accident (both on and outside planes), etc. And if you want to be precise, you should include transportation to use main transportation, so walking for a bus stop, driving to airport, etc. But is it not "commercial flight safety": there is a lot of social parameters (who flight, who must drive a lot, etc.), and so it is a bad metric for your purpose

Seems like “per mile” is still correct. Even in the risks occur in specific phases, the fact they don’t occur in other phases is what makes air travel less risky.

@300D7309EF17 Well that's my point. If you make a metric "per mile", it implies that the risk scales linearly with distance, meaning if you fly twice the miles, you have twice the risk. But, as I believe the statistics show, that is not correct for airflight. Two flights of 1000 miles each is significantly more risky than one flight of 2000 miles. If 80% of accidents (and roughly 70% of fatalities) are independent of the total distance, then "accidents/deaths per mile" seems less useful than e.g. "accidents/deaths per flight. That's what I'm asking about.

I think the reason why "deaths per passenger mile" was picked was to make comparisons with other modes of transportation fairer. The "passenger mile" measures the utility (or usefulness?) of the mode of transportation (how far it gets you), whereas the number of deaths (or accidents) measures the risk taken to achieve that utility. Of course, when travelling by car, trips on multilane highways are statistically much safer than getting to the highway on secondary roads, but does it make sense to separate the trips by these criteria when you want to compare them with air travel?

@rob74 Yes, I know, whatever metric you use, you have to use the same metric if you want to compare e.g. cars and airplanes, I acknowledged that in my question. But my point is that the metric itself does not accurately model the risk of airflight compared to cars because the risk of a car accident is, say, 95% dependant on distance (double the distance, ~double the risk, all else being equal), while for airplanes it's only 5% dependant as per the statistics I gave. That vast difference makes it an almost meaningless comparison.

""Fatalities by airplane color" does not seem useful" well, if you want to compare the statistic with cars, then colour might well be an important metric since yellow cars are statistically less involved in incidents

@MichaelHall I reworded the question based on your suggestion. Note that "comparison with other modes of transportation" was not the core of the question. The core question is rather "Is 'deaths per mile' the best metric to evaluate airflight safety when compared to other metrics, in particular 'deaths per flight'?"

@sophit That actually illustrates my point rather well. If the color of a car is relevant for car safety statistics (with a real reason behind it such as a correlation with the owner's psychology, or type of car) but not for airplane safety, then the comparison between car and airplane safety based on color is still invalid. Whatever metric you choose has to be equally (!) relevant to both plane and car, otherwise you're not comparing things ceteris paribus.

it's going to be hard to answer this with all the strawmen in the way.

@300D7309EF17 Could you be more specific what remains unclear about the question?

Given the limited duration of every human life, deaths per hour of exposure would probably be the better metric. But then it would only be fair to add the way to and from the airport and add the hours of waiting and security theater to the travel time, too. This makes the calculation overly complex.

Deaths per passenger-mile is the best we can get, and it is good enough.
It's crucial to realize that estimating the safety of a given journey is always a very crude approximation. Unless you have lots of robust statistics  for the particular trip you plan to take, you're left extrapolating from data that's weakly correlated at best.
If you're considering whether to fly from Atlanta to Seattle or take the car instead, you would ideally want to look at the fatality rate on Atlanta-Seattle flights on a particular airline (because maintenance and crew quality varies), on a particular aircraft model, in the last five years (because the industry evolves a lot), and you would want at least 10 crashes in that data set to calculate at least a rough average rate. But we don't have any data remotely close to that, because accidents are so infrequent. So you're left extrapolating from data aggregated from trips between Bangkok and Jakarta and between London and Dubai, and that's already a really crude approximation.
Compared to that approximation, the issues you mention are entirely negligible. Most commercial airline flights are not much shorter than 1 hour and not much longer than 10 hours, so in the worst case, those really long flights will dilute the take-off/landing crash rate by a factor of ten. That's not that much of a difference actually, since there's several orders of magnitude separating air travel safety from passenger cars. Similarly, there's likely several orders of magnitude separating the safest flights from the least safe, so that's the general level of accuracy you can expect when aggregating them all.
Deaths per journey aren't too useful, because that metric either wouldn't work at all (if you considered all car journeys equivalent, 5-minute trips to the mall will dilute your statistics much more than long flights) or would require looking at only car trips from Atlanta to Seattle (not enough data for statistics again).
If you give up trying to compare modes of transportation and only want to compare different flights with each other, check out Arnold Barnett: Measure for Measure ("A statistician offers his perspective on the relative usefulness of different ways of measuring aviation safety"). It thoroughly compares and discusses various ways to measure safety. Table 1 shows that "deaths per passenger-mile" is still the best predictor of mortality risk per a randomly chosen flight, although the correlation coefficient is "just" 0.57.

"That's not that much of a difference actually, since there's several orders of magnitude separating air travel safety from passenger cars." According to Wikipedia the death rate per hours is 130 for cars and 30 for airplane so not even one order of magnitude difference.

@sophit Yeah but that's just circling back to the problem of picking a reasonable metric. Deaths per hour are super skewed in favour of cars because of their much lower speeds (and hence also distance traveled). Taking the same trip will take you much shorted by air than by car, making passenger-miles (kilometres) much more relevant. Looking at the third column in your Wikipedia table, there's two orders of magnitude between air and car.

I think that the "death per pax mile" is the appropriate measure because it includes achieving the goal of getting from A to B.

Driving from the Indianapolis airport to the Atlanta, GA airport is roughly 11 hours (and involves navigating 4 major metropolitan areas).
Flying from IND to ATL is roughly 3 hours.

Even if the "death per hour" rate were exactly the same for air & ground transport, I'd be exposed nearly 4 times longer when traveling by car, thus air travel is still a safer way to go.
However, since both trips cover the same amount of ground and both achieve the same goal of getting me from Indianapolis to Atlanta, "deaths per mile" does seem to give me the most useful metric.
If we were to use "deaths per hour" as the metric, I'd have to stop somewhere around Elizabethtown, KY when driving to keep the risk comparison the same as flying, and that wouldn't achieve my goal.
If one were to use the the "deaths per trip", how do you define "trip"?

If I fly, but I have a stop in St. Louis, is that considered 2 trips?

It's certainly considered 2 cycles for the aircraft, in terms of maintenance and air frame life cycle.

If I drive, would that be considered two or even three trips?

After all, I'm not going to make this trip on one load of fuel (a generic term to cover both gas/diesel and electric powered vehicles). If I drive until my fuel is depleted, is that one trip?

The problem is, that no other metric is significantly better, and all of them favour one mode of transport over all others.
For example, cars can be considered much safer, since most people survive a road-traffic collision, even without taking into account that a large number of collisions aren't reported/recorded precisely because there is little damage as a result.
So, how do you want to measure 'safety'?
Is number of deaths even the best measure?

"that no other metric is significantly better," Could you elaborate why you think that? It's exactly what my question is about. "all of them favour one mode of transport over all others." What do you mean by that? By my reasoning, "deaths per distance" make sense for all other modes of transportation because the accident/fatality risk is about evenly distributed over the distance, which is AFAIK uniquely not the case for air travel.

The distance is a problem. For cars, do you consider just commuters? Or all road traffic (so lorries which do a lot of miles and on safer roads). But so, should you include all GA or not?

@MikeB also I don't want to get into a discussion about which negative event to consider, because that's not the point of my question. If you prefer, use "number of accidents" or "amount of property damage in USD" or whatever els per X. My question is about the X specifically in airflight, about the independent variable in that ratio.

@Hackworth  But that is the crux of my answer - you are saying that you don't like "Deaths per mile" so what DO you care about? The only perfect metric is "am I going to die/injured/other on " And even for that, you need to include getting to/from the airport somehow, and need to decide whether death/injury/close-shave matters.

@MikeB Honestly, other than your assertion that no other metric is better than "deaths per mile", you haven't answered my question. I'll repeat my request for why you think there is no better metric. And no, I don't care about the safety of getting to the airport or whatever, only about the time between getting on and off the airplane. Let's not make it more complicated than what I've asked for.

@Hackworth  Yes, I realised that too, when I was nearly finished writing it! But I think the point is that pretty much everyone else agrees that "Deaths per mile" or "Deaths per journey" are the best we have. Fundamentally, only you can decide what matters to you ie there is no such thing as a perfect metric for this (and rarely for any other query).

Deaths-Per-Flight Suggests a Perverse Strategy To Achieve Safety; Deaths-Per-Passenger-Mile Captures All Of Your Concerns
Deaths per passenger mile (D/PM) has a number of advantages to recommend it, and ultimately is superior to deaths-per-flight (D/F) for virtually all use cases, but especially the use case you want to use it for:

predictive of airtravel accidents/fatalities of the industry as a whole, not any individual flight.

Not all flights are of the same duration, nor is duration the only factor (is the flight mostly over the Atlantic Ocean, vs. a flight path exclusively over flat agricultural-use land, for example).
The good news is that, when analyzing air travel at the industry-as-a-whole scale, all of those factors end up in the same blender, reduced to a single statistical slurry: how many units failure are suffered per unit of service; how you define the 'unit of service' and how you define the 'unit of failure' really matters here.
Unit of Service
What's good about air travel? Why do we want it? If a plane flies from Boston to Los Angeles, is that a good thing?  What if the plane is empty?
Since we're talking about 'air travel' we can assume that an empty plane going from Boston to Los Angeles is of zero value, but if we define the unit of service as 'a flight' which is what 'deaths per flight' does, we end up valuing the flight from Boston to Los Angeles as 'a flight.'  If it crashes, we get a value of 2-deaths per flight (the folks up front) for that single case.  This will bring the real number of 'deaths per flight' down as a consequence.  If we, instead, value the movement of passengers, we run into a similar issue if a plane is only half-full for that same trip. A full plane produces twice as much 'travel' as a half-full plane. So the unit of measuring travel is not 'a flight' but is instead the movement of people from one place to another.  We can do this on a per-capita basis to control for varying numbers of people on various flights, and as such we end up valuing planes that are more full (because they're giving us more 'travel' value by moving more people around).  More people on a flight, however, means 'deaths per flight' will go up - because we're trying to get more butts in seats... including the seats that crash.
Similarly, a flight from Boston to Los Angeles gives us more 'travel' than a flight from Boston to Kansas City, holding passenger count constant.  The folks who went to LA got further.  Just as we expressed 'travel' on a per capita basis, we can express it as a form of distance by expressing it on a 'per mile basis.'  Travel, thus, ends up defined in units of 'Passenger Miles.'  Regardless of mode, the value of travel is a function of how many people get moved how far, more people = more passengers, longer distances = more miles.  Now I can compare a flight BOS->LAX with 150 people to a flight BOS->MCI with 120 people on board in terms of relative safety.  To the extent that both planes have the same chance to crash - as you propose - the BOS-MCI flight is giving us fewer deaths per flight - but we're getting less actual travel out of that flight.  Deaths per flight fails to capture that.  If we have the same chance to crash either way, clearly we should only want to fly when we're going further - in other words we're getting more travel value per unit of risk taken.  The good news is that by measuring travel as a passenger-mile, we don't lose that intuitive calculus: the LAX trip produces more death, but it also gives us tremendously more value in terms of passenger miles, compensating us for the added risk.
Unit of Failure
Unit of Failure is the risk we're trying to estimate. It is widely known that a plane crash will ruin your whole day, and for purposes of pessimism we can simply say that everybody who's on a plane that crashes, dies. It's not true, but it helps us avoid underestimation of the risk.  Just as we ran into trouble with trip length, however, we now have a problem that planes are varying degrees of full so a given crash isn't going to produce a consistent value for deaths.  Moreover, we defined 'unit of service' as 'passenger mile' so we need to at least convert to crashes per that unit.  This, we can do.  The problem remains, however, than a flight that has two people on it is not the same as a flight that has 200 people on it - but both are 'a crash' so the unit of failure should probably be able to capture the severity of the event, which is a function of the number of souls aboard: deaths perfectly captures the aspect of crashes that we don't like, and it's conveniently in similar units to 'passenger mile.'
Thus, deaths per passenger mile is chosen as a metric because it allows the aggregation of data from jets and turboprops and piston craft; transatlantic and regional; charter and scheduled; red-eye and prime time; deadhead and fully-booked.  That aggregation smooths out the noise from any of those specific factors - but also allows comparison between them for sensitivity analysis that could reveal, for example, if transatlantic flights were more dangerous (because crashing in the middle of the Atlantic involves a MUCH lower chance that you'll be rescued) or not (because, as you point out, cruise phase isn't where the bulk of trouble occurs).
Intermodal comparisons
Deaths per passenger mile also facilitates comparison to other modes of transportation on a normalized basis.  One flight could move 200 people, requiring a minimum of 50 4-seat passenger cars to achieve the same result. Assuming 100 miles traveled, differences would wash, but if the plane traveled more than 600 miles we now need to talk about those cars stopping for the night, or proceeding to drive while tired (resulting in higher accident rates on a per car basis) - suppose a single overnight, 100 4-seat passenger car trips are now needed to achieve the same net result of a single airline flight.  If both have a 2% chance of a given trip going badly, cars are now killing twice as many people for the exact same unit of production (200 people moved the given distance).
D/PM controls for all of that.  Let's take 1000 miles for simplicity of math, and assume an automobile driver needs two days to make the same journey.
One flight of 200 people over a distance of 1,000 miles, with a 2% rate of 'everybody dies' = 200 x 2% / 200,000 passenger miles = 0.00002 D/PM.
50 4-seat car trips moving 200 people over a distance of 500 miles per trip, so needing two trips, with a 2% rate of 'everybody dies' = 50 x 4 x 2% x 2 / 200,000 passenger miles = 0.00004 D/PM.
As you'd probably intuit, if a car trip and a plane trip have the same chance to end in tragedy - the fact that I'd have to take twice as many trips by car to achieve the same result gets reflected in the D/PM - but NOT in the 'crashes/deaths per trip' statistic.
To add one more wrinkle, automobile crashes have much less energy involved and so have a much higher chance that one or more passengers survive a severe crash than an airliner disaster does.  So a more realistic comparison is that the car trips have a 2% rate of 'serious crash' within which the average fully loaded 4-seat passenger car kills, on average, 2.75 of its 4 occupants (the rest suffer serious injury but ultimately survive).
Now we're not even comparing 'chance of dying' to 'chance of dying' if we look at 'chance of crash' as the statistic of interest, but D/PM corrects for this automatically.
In our 50x2 4-seat car trips for 1000 miles case, we expect 1 out of 50 (2%) to end in tragedy, killing 2.75 people (instead of 4). 50 x 2.75 x 2% x 2 / 200,000 passenger miles = 0.0000275 D/PM - much closer to, but still technically more dangerous than our hypothetical passenger flight.
D/F is subject to the same issues that 'chance of a crash' is in the final hypothetical: it's impossible to meaningfully understand if air travel is generally safe if I'm measuring 'success' as 'a plane flew from A to B.'  An empty plane produces a flight, but has almost no ability to contribute deaths-per-flight if it crashes, killing the two people up front and no one else.  The safety protocols likely to be recommended by a D/F metric used to evaluate the air travel industry is 'keep as many seats empty as possible so we get more flights.'  That doesn't reduce the number of people who die, it just means we now also lose more airplanes doing it.  D/PM escapes that fate because an empty plane, while it doesn't meaningfully contribute to the deaths-per part of the equation it also doesn't contribute to the 'passenger miles' part of the equation, because it's not usefully moving any people.
You get what you measure.
It is also equally important to note that if you value safety in terms of 'deaths per flight' instead of 'deaths per passenger mile' you indicate to airlines that they should pursue safety in those terms as well.
In other words: Fly more planes, with more empty seats.
The demand for travel doesn't change, however, so this produces an aviation system that kills the same number of people (same % chance that any given flight you're on kills you) but now involves far more planes.  It keeps deaths per flight down by simply ensuring that any given crash involves fewer people.  Same % of travelers killed by this strategy has several downsides: air travel costs more (need to hire more pilots per passenger, buy more planes per passenger, consume more fuel per passenger), and now we're also crashing more planes in absolute terms in order to achieve safety.
In public policy (my field) this is known as a perverse incentive structure.  Deaths per flight, as a metric, produces more waste and as long as those planes are hitting at least one person on the ground, we're also getting an actual increase in death.
Measuring deaths per passenger mile, however, indicates the optimization strategy of: If we're going to risk a plane crash, let's make sure we're moving as many people as we can, as far as we can, so that we get the maximum value per crash we have to suffer.  Yeah, that means more people die per plane crash - but for each of those deaths, the system as a whole is producing more travel which is the thing we actually want.  The empty airplane flying BOS->LAX illustrates that it's not actually 'flights' we want. Nor is it 'airplanes.'  What we ask of the system is 'move people a distance.'
So D/PM is optimized by filling large, efficient aircraft packed to the gills with the best safety equipment we can, operated by highly trained professionals who can be selectively chosen because we need comparatively fewer of them.
It's also optimized by preferring that flights operate in their safest mode for as much of the flight as possible. As you note, takeoff/landing are where the danger is: so for any given takeoff/landing we want to get as many passenger miles as we can.  In other words, we want to use air travel for longer-distances, where they can justify the comparatively high risk on a per-flight basis - exactly the situation where the 'per trip' risk is amplified by limits of the passenger car, because one flight can replace many car trips for the same passenger miles value. (This is ignoring the fact that, because of the amateur nature of its operator population, car travel is absurdly more dangerous.)
What actually predicts accidents?
Because we're measuring aggregate system performance for a system that is used beyond a de minimis level, predicting 'will there be a plane crash' is trivially easy: Absolutely, there will be.  Either metric tells you this, because there's enough planes flying around that it's not a question of 'if' anymore, it's a question of 'how many.'
Deaths-per-flight measures how many people are, on average, involved in a typical plane crash.  All it tells you is how many people are in the plane.
Deaths-per-passenger mile measures how many people are killed as a function of the satisfaction of the purpose those planes fly for. This has the added value of allowing you to predict how many people will die if you have an estimate of the total distance people will need to travel by air.  If you want to know how many plane crashes that will involve, you do still need to come up with an average human-beings-per-flight but the use cases for 'how many crashes' are few and far between (anticipated need for accident investigators is one potential case, however).

The answer to your specific question between those two metrics is: No
No, "death per passenger mile" is not a correct metric compared to "death per flight". Why is that? Well, at leat two reasons justify this:

First, dying at the end or the beginning of my journey from Canada to Australia does not matter a lot: in the end, I'm dead. But in one case, if the airplane crashes on landing, the death per passenger mile would be significantly better than in the case of a plane crashing during take-off. On the other hand, given that many accidents occur during takeoff, you could say in an hypothesis that all airplanes have accidents, that the ratio is X deaths to 0, which does not make sense.
Second, the passenger mile number changes considering how many people are inside the plane: but whatever the number of bookings in the plane, you do not want it to crash anyway (unless it is a freighter plane where one might consider that safety rules could be lighter, and even that is not the case AFAIK)
Third, since most accidents of airplanes' causes are not linked to the length of the trip, you should note take it into account. On the other hand as said above, one flight is dangerous because it passes through some specific moments such as takeoff*

*Please note that I'm not definitely stating that take-off is very dangerous nor is the most dangerous part of a flight: I am not entirely sure of that. I'm just taking it as a plausible example for this answer.

If you want to get rid of any dependency on length, what about death per journey or per hour? That should give a "more correct" number according to your assumptions. Those statistics should also be readily available.
I agree anyway with @MikeB that "no other metric is significantly better, and all of them favour one mode of transport over all others". For example, if you consider the death per journey as I suggested, you have anyway somehow to consider that a typical journey via airplane would be much longer than a typical journey by car... Maybe you could divide the death per journey via the typical journey's length.

Death per hour does nt take into account the frequency of journey between many cars a day and one plane a day (for example)
If you consider "per journey of travel", then a ship will not be safe because it travels long and slow compared to short and fast car traffic in campaign, where many fatalities occur

Deaths per hour would probably be the better metric. But then it would only be fair to add the way to and from the airport and add the hours of waiting and security theater to the travel time, too. This makes the calculation overly complex.


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Hypothesis: swept back propeller would reduce drag and increase thrust
Hypothesis: swept back propeller would reduce drag and increase thrust. Any experience with this?

@RalphJ  Let's not close this question.  OP is a new contributor and the question asked was not specifically about scimitar propellers.  Although OP was referred to that question and the answers therein, he may not have had knowledge of scimitar propellers, or have even thought of searching for them.  My reference to the already answered question was as a courtesy given his question about swept-back propellers.  In my view, closing the question is inappropriate given these circumstances: a) he is a new contributor, and b) may not have known of scimitar propellers.

@ThomasPerry Closing a question as a duplicate isn't a negative for the OP, since not every dupe is easily found in a search. The point of closing duplicate questions isn't as a "down" for the user asking the question, but rather to keep the forum organized. And, secondarily, to keep all (or at least as many as possible) answers to a given question in one place, the original question, rather than having some answers posted to the first place a question was asked, and others posted to other threads. Neither "new contributor" nor "didn't know about" plays into these goals.

@RalphJ  Excellent, & quite understandable.  Thank you for the succinct explanation.

An interesting question! There are scimitar bladed propellers.  This se-aviation question has two answers giving an excellent brief summary of the advantages of these props, and also their disadvantages or limitations.  In general, the advantages are higher tip speeds and reduced shock, which reduces drag, as well as net slightly smaller prop diameter, compared to straight bladed props, which reduces vibration and noise.  The curved shape of the blade provides a small increase in prop efficiency.  The disadvantages relate to torsion, or twisting, of the blades due to the distribution of thrust along the length of the blade. This twisting tendency is manageable as long as the sweep back of the blade is gentle and not excessive.
Scimitar propellers have been around for a long time.  They are in common use on many general aviation aircraft.


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Does propeller width have a material impact on thrust?
I have seen propeller thrust calculations and they do not appear to have the variable of propeller width. They only use diameter or radius. What if any impact does width of the propeller have?
Here is one such post with calculation.
Is this equation for propeller thrust correct?

The short answer is yes, it has to.  I would presume that one of the other variables related to the coefficient of "lift' of the prop airfoil would include the width.  It might help if you included the formula you are using.

Agree with Michael Hall.  A propeller is essentially a spinning wing, producing lift.  Area is in the lift equation, although, just as in wings, a lower aspect propeller may have a higher L/D ratio, but may be able to absorb more power from the engine to create greater thrust.  It would be an interesting study comparing a wider 2 blade, a 4 blade of the same (total) area, and 2 contrarotating 2 blades.

By width you mean the chord or the thickness of the blade?

Not sure what you mean with "width" but for sure camber is a major factor, like the wing camber on an aircraft wing.

Usually when informed that an ambiguous question has been asked, people here clarify in order to avoid misunderstanding. You should too.

Yes, the average blade width -- called 'solidity' for a helicopter rotor and 'activity factor' for a propeller is an important parameter for blade thrust.
However, it gets boiled into the thrust and power coefficients typically used when evaluating propeller and aircraft performance.
Just as the chord is important for a wing, but it does not appear in the formulas for lift and drag coefficient.

I believe the Bell "Huey" helicopter had longer chord blades and the auto rotating pilots may have been glad of it.

I found the following explanation of lift that does have a variable for airfoil area which would include the width of the airfoil.

https://www.experimentalaircraft.info/flight-planning/aircraft-lift-formula.php